Statistical mechanics approach to sparse noise denoising

Reconstruction fidelity of sparse signals contaminated by sparse noise is considered. Statistical mechanics inspired tools are used to show that the ℓ
₁
-norm based convex optimization algorithm exhibits a phase transition between the possibility of perfect and imperfect reconstruction. Conditions characterizing this threshold are derived and the mean square error of the estimate is obtained for the case when perfect reconstruction is not possible. Detailed calculations are provided to expose the mathematical tools to a wide audience.